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Asymptotic enumeration of

labelled 4-regular planar graphs

Marc Noy ∗ Clement Requile † Juanjo Rue ‡

January 20, 2020

Abstract

Building on previous work by the present authors [Proc. London Math.Soc. 119(2):358–378, 2019], we obtain a precise asymptotic estimate forthe number gn of labelled 4-regular planar graphs. Our estimate is of theform gn ∼ g · n−7/2ρ−nn!, where g > 0 is a constant and ρ ≈ 0.24377 isthe radius of convergence of the generating function

∑n≥0 gnx

n/n!, andconforms to the universal pattern obtained previously in the enumerationof planar graphs. In addition to analytic methods, our solution needsintensive use of computer algebra in order to work with large systemsof polynomials equations. In particular, we use evaluation and Lagrangeinterpolation in order to compute resultants of multivariate polynomials.We also obtain asymptotic estimates for the number of 2- and 3-connected4-regular planar graphs, and for the number of 4-regular simple maps,both connected and 2-connected.

1 Introduction and statement of results

In a previous work [9], the authors solved the problem of enumeratinglabelled 4-regular planar graphs. The solution was based on decompos-ing a 4-regular planar graph along 3-connected components and findingequations relating the generating functions associated to several classesof planar graphs and maps. This produced large systems of polynomialequations from which counting coefficients could be extracted using Maple.However, the size of the systems and the complexity of the equationsinvolved prevented a direct elimination approach from which to obtainasymptotic estimates of the coefficients.

∗Universitat Politecnica de Catalunya, Department of Mathematics. E-mail:marc.noy@upc.edu. Supported by grants MTM2017-82166-P and MDM-2014-0445.

†Technische Universitat Wien, Institute for Discrete Mathematics and Geometry, Austria.E-mail: clement.requile@tuwien.ac.at. Supported by the Special Research Program F50-02Algorithmic and Enumerative Combinatorics of the Austrian Science Fund.

‡Universitat Politecnica de Catalunya, Department of Mathematics. E-mail:juan.jose.rue@upc.edu. Supported by grants MTM2017-82166-P and MDM-2014-0445.

1

In the present paper we reconsider the problem and are able to obtainsingle polynomial equations defining the various generating functions. Al-though these equations are in some cases very large in terms of the degreeand the size of the coefficients (two of them would need about 30 pageseach to be printed), we are able to compute their Puiseux expansions atthe dominant singularities and deduce precise asymptotic estimates. Theestimates follow the universal pattern obtained previously in the enumer-ation of planar graphs and maps [2, 1, 5, 8].

We first find the equations for 3-connected 4-regular planar mapscounted according to simple and double edges. We could not succeedwith direct elimination schemes due to the enormous size of the equa-tions involved, which seem to require too much space even for a relativelypowerful computer. Instead we use evaluation and Lagrange multivari-ate interpolation to compute the associated Sylvester resultant. In orderto guarantee correctness of our results, we need an upper bound on thedegree of the resultant. In our case the best upper bound we obtain is160, hence we have to interpolate at 161 points. This computation tookseveral hours of CPU and allowed us to compute the minimal polynomialsof the generating functions T1(u, v) and T2(u, v) of 3-connected 4-regularplanar maps, as shown in Section A.2 of the Appendix.

Once the equations for the Ti(u, v) are obtained explicitly, furtherelimination gives the minimal polynomial P (x, y) for the generating func-tion C•(x) = xC′(x) of vertex-rooted connected 4-regular planar graphs.Since the polynomial P is too large to be displayed in print, we provide alink to a fully annotated Maple file1, where all our computations can bereproduced. In the sequel we refer to these file as “the Maple sessions”.From P we compute the dominant singularity of C(x), which turns outto be an algebraic number of degree 14, as shown in Theorem 1. Then weperform similar computations for 2-connected and 3-connected 4-regularplanar maps, as well as for 4-regular simple maps (in this last case theminimal polynomials are small enough to be reproduced in Section A.3 ofthe Appendix). These results, together with the corresponding asymptoticestimates, are the content of the following three theorems.

First, our main result is an estimate for the number of connected andarbitrary 4-regular planar graphs.

Theorem 1. (a) The number cn of connected 4-regular labelled planargraphs is asymptotically

cn ∼ c · n−7/2 · γn · n!, with c ≈ 0.00139114 and γ = ρ−1 ≈ 4.10228,

1https://dmg.tuwien.ac.at/crequile/4-regular_planar_maple.zip

2

where ρ ≈ 0.24377 is the smallest positive root of

12397455648000x14 + 99179645184000x13 − 263210377713408x11

+ 4123379191922784x10 − 1230249287613888x9

− 18655766288483533x8 + 51831438989552290x7

+ 97598878903661028x6 + 620596059256280x5

+ 15894289357702528x4 − 63729042783408384x3

− 66418928650596352x2 + 64476004593270784x

− 109267739753840648 = 0.

(1)

(b) The number gn of 4-regular labelled planar graphs is asymptotically

gn ∼ g · n−7/2 · γn · n!,

where g ≈ 0.00139116 and where γ is as in Theorem 1.

Remark. From the previous theorem it follows that the probabilitythat a random 4-regular labelled planar graphs is connected tends to c/g ≈0.99993 as n → ∞.

We can also estimate the number of 3- and 2-connected 4-regular pla-nar graphs.

Theorem 2. (a) The number tn of 3-connected 4-regular labelled planargraphs is asymptotically

tn ∼n→∞

t · n−7/2 · (γ3)n · n!,with t ≈ 0.0012070 and γ3 = τ−1 ≈ 4.08978,

where τ = 88−12√

21135

≈ 0.24451.

(b) The number bn of 2-connected 4-regular labelled planar graphs isasymptotically

bn ∼n→∞

b·n−7/2·(γ2)n·n!,with b ≈ 0.0000575832 and γ2 = β−1 ≈ 4.10175,

where β ≈ 0.2437981094 is the smallest positive root of

12397455648000x11 + 24794911296000x10 − 148769467776000x9

+ 1125304654862592x8 − 451035134375328x7

− 7923244598779392x6 + 38505114557935859x5

− 67113688868067728x4 + 70322996382137760x3

− 43445179814077952x2 + 12857755940483072x

− 1365846746923008 = 0.

(2)

Our last result is on the number of simple 4-regular maps.

Theorem 3. (a) The number un of 4-regular simple maps is asymptoti-cally

un ∼ s · n−5/2 · σ−n,with s ≈ 0.016360 and σ−1 ≈ 4.13146,

3

where σ ≈ 0.24204 is the smallest positive root of

432x8+448x7−852x6+588x5−72x4−504x3+135x2+108x−27 = 0. (3)

(b) The number hn of 2-connected (i.e. non-separable) 4-regular simplemaps is asymptotically

hn ∼ h · n−5/2 · η−n,with h ≈ 0.014477 and η−1 ≈ 4.122915,

where η ≈ 0.24255 is the smallest positive root of

108x6 + 4x5 − 136x4 + 344x3 − 425x2 + 196x− 27 = 0. (4)

Remark. All the constants in the previous theorems with one exceptionare given by explicit equations. We provide numerical approximationswith five decimal digits but they can be approximated to any desiredprecision. The exception is constant g in Theorem 1, whose approximatevalue is estimated from the first coefficients gn as discussed at the endof the proof of Theorem 1. In addition, all the polynomials shown areirreducible and their integer coefficients have no common factor.

The rest of the paper is organized as follows. In Section 2 we recall thebasic definitions on planar graphs and maps, then on algebraic generatingfunctions. In Sections 3 and 4 we recall first the various combinatorialobjects introduced in [9] and the equations satisfied by the associatedgenerated functions. Then by elimination we find the minimal polynomialsof quadrangulations and 3-connected 4-regular maps. In Section 5 we usethe results of the previous section to compute minimal polynomials for4-regular planar graphs and maps. Finally in Section 6, after providingan analytic lemma, we obtain the asymptotic estimates for all the graphsand maps of interest.

2 Preliminaries

2.1 Planar graphs and maps

Throughout the paper graphs are labelled and maps are unlabelled. Agraph is planar if it admits an embedding on the plane without edge-crossings. A planar map is an embedding of a planar multigraph up toorientation preserving homeomorphisms of the sphere. It is simple if theunderlying graph is simple. A planar map M is always considered rooted:an edge ab of M is distinguished and given a direction from a to b. Thevertex a is the root vertex and the face on the right of ab as the root face.Any other face is called an inner face of M . Vertices incident with theroot face are called external vertices.

A map in which every vertex (resp. face) has degree four is saidto be 4-regular (resp. a quadrangulation). By duality, quadrangulationsare in bijection with 4-regular maps. Notice that quadrangulations canhave“degenerate” faces consisting of a double edge with an isthmus in-side. A quadrangulation with at least eight vertices is irreducible if every

4

4-cycle forms the boundary of a face. Irreducible quadrangulations areknown to be in bijection with 3-connected maps (see [1]).

The following concepts are taken from [9] (see also [7]). A diagonal ina quadrangulation is a path of length two whose endpoints are externaland the central point is internal. If uv is the root edge, then there aretwo kinds of diagonals, those incident with u and those incident with v.By planarity both cannot be present at the same time. A vertex of degreetwo in a quadrangulation is called isolated if it is not adjacent to anothervertex of degree two. An isolated vertex of degree two will be called a2-vertex. By duality, a 2-vertex becomes (in the corresponding 4-regularmap) a face of degree two not incident with another face of degree two.We call it a 2-face. Furthermore, we say that an edge is in a 2-face if itis on its boundary, and ordinary otherwise. Note that since the numberof edges of a 4-regular map is even, so is the number of ordinary edges.

2.2 Algebraic generating functions

A power series f(x) is algebraic if it satisfies a polynomial equation of theform

P (f(x), x) = pk(x)f(x)k + pk−1(x)f(x)

k−1 + · · ·+ p1(x)f(x)+ p0(x) = 0,

where the pi’s are polynomials in x. If the polynomial P (y, x) is irreduciblethen it is unique and is called the minimal polynomial of f(x). An alge-braic power series f(x) with non-negative coefficients is represented as abranch of its minimal polynomial P (y, x) in the positive quadrant passingthrough the origin. This last condition represents the fact that there isno graph with an empty vertex set. We call this branch the combinatorialbranch. It defines an analytic function in a disk centered at the originwith positive radius of convergence ρ. Since the coefficients of f(x) arenon-negative,it holds by Pringsheim’s theorem [4, Theorem IV.6] that ρ isa singularity of f(x), called the dominant singularity. In this paper ρ willalways be a branch-point (f(ρ), ρ) of P (y, x), that is, one of the commonroots of

∂P

∂y(y, x) = 0, P (y, x) = 0,

Equivalently, it is the smallest positive root of a factor of the discrimi-nant of P (y, x) with respect to y (see [4, Section VII.7]). All the algebraicpower series f(x) in this paper admit a Puiseux expansion as x → ρ− (i.e.|x| < ρ and x → ρ) of the form

f(x) = f0 + f2

(1− x

ρ

)+ f3

(1− x

ρ

)3/2

+O

((1− x

ρ

)2),

which is the local expansion of f(x) near ρ− corresponding to the com-binatorial branch of P (y, x). Furthermore, f0, f2 and f3 are algebraicconstants and f3 > 0.

Using the Newton polygon algorithm (see [6, Section 6.3]), one cancompute this expansion in time polynomial in the degree of f(x). Thisalgorithm has been implemented as the function puiseux in the Maple

package algcurves and this is what we use in Section 6.

5

3 Equations for quadrangulations

We follow the combinatorial scheme introduced in [9, Section 2], which wesummarize here. Following Mullin and Schellenberg in [7], we partitionsimple quadrangulations into three families:

(S1) The quadrangulation consisting of a single quadrangle.

(S2) Quadrangulations containing a diagonal incident with the root ver-tex. By symmetry, they are in bijection with quadrangulations con-taining a diagonal not incident with the root vertex. Each of thosetwo classes can be partitioned into three sub-classes Ni for i = 0, 1, 2,according to the number i of external 2-vertices.

(S3) Quadrangulations obtained from an irreducible quadrangulation bypossibly replacing each internal face with a simple quadrangulation.We denote this family by R.

We use simple quadrangulations to obtain generating functions forgeneral quadrangulations encoding 2-vertices [9, Section 2.2]. This is donein two steps. First, we obtain equations for quadrangulations of the 2-cycle(see [9, Lemma 2.3]). We denote byA = A0∪A1 the quadrangulations of a2-cycle, where A1 are those whose root vertex is a 2-vertex (by symmetry,they are in bijection with those in which the other external vertex is a2-vertex), and A0 are those without external 2-vertices.

Finally we obtain equations for arbitrary quadrangulations B. Wedecompose B = B0 ∪ B∗

0 ∪ B1, where B1 are those in which the rootedge is incident to exactly one 2-vertex, and B0 ∪ B∗

0 are those in whichthe root edge is not incident to a 2-vertex. Furthermore, B∗

0 are thequadrangulations obtained by replacing one of the two edges incident withthe root edge in the single quadrangle.

Irreducible quadrangulations. We use [1, Equation (9)]. Let sn bethe number of irreducible quadrangulations with n inner faces. Then theassociated generating function S(y) =

∑n≥0 sny

n verifies the followingimplicit system of rational equations

S(y) =2y

1 + y− y − U(y)2

y(1 + 2U(y))3, U(y) = y(1 + U(y))2.

By eliminating U(y) from the above system and factorising we obtain apolynomial satisfied by S(y). Then by expanding the roots of each factorin series of y near zero, one can check that the minimal polynomial ofR(y) is given by

PS(S(y), y) = (y5 + 8y4 + 25y3 + 38y2 + 28y + 8)S(y)2

+ (2y6 + 12y5 + 20y4 + 10y3 − 5y2 − 4y + 1)S(y)

+ (y7 + 4y6 − y5) = 0.

(5)

From irreducible quadrangulations to simple quadrangula-tions. We use variables s and t to mark inner faces and 2-vertices, re-spectively. We write Ni = Ni(s, t) (i = 0, 1, 2) for the generating function

6

of the subclass Ni counting both the number of inner faces and 2-vertices.We denote by R = R(s, t) the generating function associated to R. Byrewriting the system (1) in [9] using these variables we get:

y = s+ 2N + S(y),

t2N = t2N0 + tN1 +N2,

tN0 = (N +R)(t(N +R +N0) +

N1

2

),

N1 = 2st(N +R+N0 +

N1

2

),

N2 = s2t3 + st(N1

2+N2

),

(6)

where y is a function of s and t.

From simple quadrangulations to general quadrangulations.We use z and w to mark inner faces and 2-vertices, respectively. In thefollowing system of equations, variables s and t are considered as functionsof z and w. As in (6), we write Ni = Ni(s, t) (i = 0, 1, 2). We denote byAj = Aj(s, t) the generating function of the family Aj (j = 0, 1).

The equations in this context are as follows; see the details in [9],in particular the topmost equation in page 365, Equation (2) and theequations in Lemmas 2.3 and 2.4:

s = z(1 + A)2,

t(1 + A)2 = w + 2A+ A2,

Q0 = s(2N0 +N1 + S(y)) + (2A+ A2)Q1,tQ1 = N1 + 2N2,

E = z(1 + A)4 − 4zA2 + 4zwA2,

wA = wA0 + 2A1,

wA = wA0 + (1 +w)A1,

A0 = 2zA(1 + A)

+z(Q0 +Q1 + E + 2z(w − 1)A+ 2z(1− w)A2),

A1 = zw(1 + A).

(7)

Then (see Lemma 2.5. in [9]) we have that

B0 = 2z(1 + A)(1 + A− A1) + z(Q0 + E − 2zwA2 − 2zA), (8)

B1 = 2z(1 + A)A1 + zw(Q1 + 2zA2), (9)

B∗0 = 2z2A. (10)

We define next three systems of algebraic equations:

S0 = (6) ∪ (7) ∪ (8), S1 = (6) ∪ (7) ∪ (9), S∗0 = (6) ∪ (7) ∪ (10).

Each of these systems is composed of sixteen equations, eighteen variables,and is strongly connected. By algebraic elimination (using the Maple

function Grobner bases), one can obtain a unique polynomial equationin any three chosen variables. We obtain the following three polynomialswhich are respectively of degree 2 in B0, B1 and B∗

0

PB0(B0, z, w) = p0,0(z, w) + p1,0(z, w)B0 + p2,0(z, w)B2

0 , (11)

PB1(B1, z, w) = p0,1(z, w) + p1,1(z, w)B1 + p2,1(z, w)B2

1 , (12)

PB∗

0(B∗

0 , z, w) = p0,2(z, w) + p1,2(z, w)B∗0 + p2,2(z, w)(B∗

0)2, (13)

7

where each coefficient pi,j(z, w), p∗i,j(z, w) is a bivariate polynomial givenin Appendix A.1. These polynomials are obtained after the eliminationprocess by factoring the resulting polynomials and choosing the right fac-tor in each case (in all cases we get only one candidate with non-negativeinteger coefficients).

4 Equations for 3-connected 4-regular maps

We shortly describe how to obtain a combinatorial decomposition schemein order to deduce equations for 3-connected 4-regular maps. More detailsare given in [9, Section 3].

The class M of 4-regular maps can be decomposed intoM0∪M∗0∪M1,

where M0∪M∗0 are 4-regular maps in which the root edge is not incident

with a 2-face, M1 are those for which the root edge is incident with exactlyone 2-face, and M∗

0 are maps in which the root is one of the outer edgesof a triple edge. These classes are in bijection with the classes B0, B∗

0

and B1 from the previous section, as the dual of a quadrangulation withℓ 2-vertices is a 4-regular map with ℓ 2-faces.

We denote by M0(q, w) = M0, M1(q, w) = M1 and M∗1 (q, w) = M∗

0

the associated generating functions, where variables w and q mark 2-facesand ordinary edges, respectively. Observe that when setting w = q, onerecovers the enumeration of 4-regular maps according to half the numberof edges. Due to the bijection with quadrangulations it follows that

M0(q, w) = B0(q, w/q),M1(q, w) = B1(q, w/q),M∗

0 (q, w) = B∗0 (q, w/q).

(14)

The next step is to decompose the previous classes. Given a map M ,let M− be the map obtained by removing the root edge st, whose endointss, t are called the poles of M . As shown in [9, Lemma 3.1] we have

M0 = L ∪ S0 ∪ P0 ∪ H, M1 = S1 ∪ P1 ∪ F ∪ F ,

where

• L are maps in which the root-edge is a loop.

• S = S0 ∪ S1 are series maps: M− is connected and there is an edgein M− that separates the poles. The index i = 0, 1 refers to thenumber of 2-faces incident with the root edge.

• P = P0∪P1 are parallel maps: M− is connected, there is no edge inM− separating the poles, and either st is an edge ofM− or M−{s, t}is disconnected. The index i = 0, 1 has the same meaning as in theprevious class.

• H are polyhedral maps: they are obtained by considering a 3-connected4-regular map C (called the core) rooted at a simple edge and pos-sibly replacing every non-root edge of C with a map in M.

• F (resp. F) are maps M such that the face to the right (resp. tothe left) of the root-edge is a 2-face, and such that M − {s, t} isconnected.

8

Let L, S0, S1, P0, P1, F and F be the generating functions of thecorresponding families each in terms of the variables q and w. In the caseof S1 and P1, we count the number of 2-faces minus one (instead of thetotal number of 2-faces) and the number of ordinary edges plus two. Inboth F and F we count the number of 2-faces minus one and the numberof ordinary edges (and in particular F = F by symmetry).

Finally, T1 and T2 are the classes of 3-connected 4-regular maps rootedat a simple and at a double edge, respectively. We denote by T1(u, v) andT2(u, v) the corresponding generating functions, where u and v respec-tively mark ordinary edges and 2-faces. The main purpose of this sectionis to obtain the minimal polynomials for both T1 and T2.

The system for T1(u, v). The following is the system (5) from [9,Lemma 3.2]. We include Equations (11)-(13) which define implicitly M0,M1 and M∗

0 in terms of q and w.

(1 +D)H = T1(u, v),u = q(1 +D)2,v = w + q(2D +D2 + F ),

M0 = S0 + P0 + L+H,qM1 = w(S1 + P1 + 2qF ),M∗

0 = 2q2D,L = 2q(1 +D − L) + L(w + q)L,S0 = D(D − S0 − S1)− L2/2,S1 = L2/2,P0 = q2(1 +D +D2 +D3) + 2qDF,P1 = 2q2D2,0 = PB0

(M0, q, w/q),0 = PB1

(M1, q, w/q),0 = PB∗

0(M∗

0 , q, w/q).

(15)

Observe that all the generating functions (including u and v) are functionsof q and w.

The system for T2(u, v). For maps in H the root of the core is asimple edge, ence we need to modify slightly the system of equations (15)in order to get an equation for T2(u, v). We adapt [9, Lemma 4.1] to themap setting and obtain

F = S2 ∪H2,

where S2 are networks in F such that after removing the two poles thereis a cut vertex, while H2 are networks in F whose root edge is incident toa 2-face. We denote by S2 and H2 the corresponding generating functions.

Equations for F , S2 and H2 are deduced in [9, Equation (7)]. Com-bining them with the decomposition explained for T1 we get the followingsystem of equations. We observe that here the minimal polynomial of B0

it is not needed. Similarly to (15), all the generating functions involved

9

in (16) depend on q and w but we omit to write the arguments.

vH2 = T2(u, v),u = q(1 +D)2,v = w + q(2D +D2) + F,

qM1 = w(S1 + P1 + 2qF ),M∗

0 = 2q2D,L = 2q(1 +D − L) + (w + q)L,S0 = D(D − S0 − S1)− L2/2,S1 = L2/2,P0 = q2(1 +D +D2 +D3) + 2qDF,P1 = 2q2D2,F = S2 +H2,S2 = (w + q(2D +D2) + F )(w + q(2D +D2) + F − S2),0 = PB1

(M1, q, w/q),0 = PB∗

0(M∗

0 , q, w/q).

(16)

4.1 The minimal polynomials of T1 and T2

The next step is to compute the minimal polynomials PT1(T1, u, v) and

PT2(T2, u, v) defining implicitly T1 and T2 as functions of u and v. In

what follows, we present the method used to obtain PT1, which is based

on evaluation and (Lagrange) interpolation. The same method is thenused to compute PT2

.

Evaluation and interpolation for PT1. First, from (15) one can

eliminate variables M0, M1, M∗0 , S0, S1, P0, P1, D, H and F so that only

the following irreducible polynomial equations remain:

QT1(T1, q, w) = 0, Qu(u, q, w) = 0, Qv(v, q, w) = 0.

Notice that these equations define implicitly T1, u and v as functions of qand w. From there we compute the resultant of Qu and Qv with respectto w and find its unique combinatorial factor Q1(u, v, q), that is, the onewhose Taylor expansion at q = 0 has non-negative integer coefficients. Thepolynomial Q1 has degree 10 in both u and v, and 16 in q. We computesimilarly Q2(u, T1, q), the unique combinatorial factor of the resultant ofQT1

and Qv with respect to w. It has degree 10 in u, 20 in T1, and 16 inq. This gives the system:

Q1(u, v, q) = 0, Q2(T1, u, q) = 0.

If we could now compute directly the resultant of Q1 and Q2 withrespect to q, this would lead to a polynomial equation R(T1, u, v) = 0having PT1

as one of its factors, and we would be done. The classicalresultant algorithm creates a Sylvester matrix of size 32, the sum of thedegrees of q in Q1 and Q2. In our case both polynomials are dense and thecoefficients of the matrix are bivariate polynomials in Z[u, v] and Z[u, T1],respectively. This computation seems to require too much space even for arelatively large computer. Instead we proceed indirectly using evaluationand interpolation.

10

For any fixed and sufficiently small integer k > 0, we can computeeffectively the combinatorial factor Fk(T1, u) of the resultant of Q1(u, k, q)and Q2(T1, u, q) with respect to q. The degree of v in PT1

is yet unknown,but let us suppose that it is d. If we repeat this computation for d + 1different values k0, . . . , kd we obtain d + 1 polynomials Fk0

, . . . , Fkdin

Z[T1, u]. The Lagrange interpolation method yields a unique polynomialin Z[T1, u][v] ≃ Z[T1, u, v], of degree d in u, and interpolating the d + 1points. This polynomial is indeed the resultant of Q1 and Q2, as allcoefficients involved are integers and because Q1 has no monomial whichcancels when evaluating v = ki, for each i = 0, . . . , d.

We now derive an effective upper-bound for d. As PT1can be obtained

as a factor of the resultant of Q1 and Q2 with respect to q, its degree in vis at most 160. Indeed, consider the associated Sylvester matrix of size 32.By construction, the coefficients in the first 16 columns are polynomialsin Z[u, v], each of degree at most 10 in v, while in the rest of the columnsthey are polynomials in Z[T1, u]. Every monomial of the resultant is hencea product of 32 bivariate polynomials, exactly 16 of which contain a termin v and of degree at most 10.

We evaluate on 161 points and then interpolate to recover the resultantof Q1 and Q2 and compute its unique combinatorial factor PT1

. Afterevaluation and interpolation with these many values2, we obtain PT1

,which has degree 8 in T1, 16 in u and 8 in v

PT1(T1(u, v), u, v) =

8∑

i=0

ti,1(u, v) · T1(u, v)i, (17)

where the ti,1(u, v) are polynomials in u and v given in Appendix A.2.

Evaluation and interpolation for PT2. We proceed similarly to

PT1. First, we eliminate from (16) to obtain an equation QT2

(T2, q, w) = 0that defines T2 as a function of q and w. Notice that the equations for uand for v are exactly the same in both systems (15) and (16), so that thetwo equations Qu = 0 and Qv = 0 are still valid in this case, as well asthe combinatorial factor Q1 of their resultant with respect to w.

Then we compute Q3, the unique combinatorial factor of the resultantof QT2

and Qv with respect to w. It has degree 10 in T2, 27 in v and 16in q just as Q2, so the same upper bound for the degree of u in PT2

holds.Finally we compute PT2

, the unique combinatorial factor of the resultantof Q1 and Q3 with respect to q, by evaluation and interpolation at 161values of u. It has degre 8 in T2, 8 in u and 13 in v and is given by

PT2(T2(u, v), u, v) =

8∑

i=0

ti,2(u, v) · T2(u, v)i, (18)

where the ti,2(u, v) is are given in Appendix A.2.

2 The algorithms for T1 and T2 take respectively around 14 and 13 hours in Maple 2018 ona personal computer (8Go DDR4 RAM, Intel (R) Core (TM) i57260U CPU @ 2.20GHz×4), byusing the libary CurveFitting and the function PolynomialInterpolation. Both are includedin the accompanying Maple sessions.

11

5 Counting 4-regular planar graphs and

simple maps

The final step is to adapt the equations introduced in the previous sectionfor graphs instead of maps. We follow the definitions and notation of [9,Section 4].

A network is a connected 4-regular multigraph G with an ordered pairof adjacent vertices (s, t) such that the graph obtained by removing theedge st is simple. Vertices s and t are called the poles of the network. Wedefine several classes of networks, similar to the classes of maps introducedin the previous section. We use the same letters, but they now representclasses of labelled graphs instead of maps. No confusion should arise sincein this section we deal only with graphs.

• D is the class of all networks.

• L,S ,P correspond as before to loop, series and parallel networks.We do not need to distinguish between S0 and S1 and between P0

and P1.

• F is the class of networks in which the root edge has multiplicityexactly two and removing the poles does not disconnect the graph.

• S2 are networks in F such that after removing the two poles thereis a cut vertex.

• H = H1 ∪ H2 are h-networks: in H1 the root edge is simple and inH2 it is double.

The generating functions of networks are of the exponential type in thevariable x marking vertices. We use letters D, L, S, P , F , H1 and H2 todenote the EGFs associated to the corresponding network class.

We next define the generating functions T (i)(x, u, v) of 3-connected4-regular planar multigraphs rooted at a directed edge, where i = 1, 2indicates the multiplicity of the root, x marks vertices and u, v mark,respectively, half the number of simple edges and double edges. Theyare easily obtained from the generating functions of 3-connected 4-regularmaps computed in the previous section, as follows:

T (i)(x, u, v) =1

2Ti(u

2x, vx), i ∈ {1, 2}. (19)

where the division by two encodes the choice of the root face.

5.1 Connected 4-regular planar graphs

The following equations are from [9, Lemma 4.2]. We denote by C•(x) =xC′(x) the exponential generating function of connected 4-regular planar

12

graphs rooted at a vertex.

4C• = D − L− L2 − F − x2D2/2,D = L+ S + P +H1 + F,L = x

2(D − L),

S = (D − S)D,P = x2

(D2/2 +D3/6

)+ FD,

F = S2 +H2,xS2 = x2v(x2v − S2),

uH1 = T (1),

vH2 = T (2),u = 1 +D,

2x2v = x2(2D +D2) + 2F,

0 = PT1

(12T (1), u2x, vx

),

0 = PT2

(12T (2), u2x, vx

).

(20)

5.2 2-connected 4-regular planar graphs

Equations for 2-connected 4-regular planar graphs are very similar, theyonly differ in the fact that networks rooted at a loop will not appear inthe recursive decomposition of a graph. Hence, we just need to removenetworks in L from the equations. Let B•(x) = xB′(x) be the EGF of 2-connected 4-regular planar graphs rooted at a vertex, where again variablex marks vertices. The system of equations defining B•(x) is given by

4B• = (D − F )− x2D(x)2/2,D = S + P +H1 + F,S = (D − S)D,P = x2

(D2/2 +D3/6

)+ FD,

F = S2 +H2,xS2 = x2v(x2v − S2),

uH1 = T (1),

vH2 = T (2),u = 1 +D,

2x2v = x2(2D +D2) + 2F,

0 = PT1

(12T (1), u2x, vx

),

0 = PT2

(12T (2), u2x, vx

).

(21)

5.3 Simple 4-regular planar maps

Let M(x) be the (ordinary) generating function of 4-regular simple maps,where the variable x marks vertices (vertices in maps are unlabelled). As

13

shown in [9, Lemma 5.1], M(x) satisfies the following system of equations:

M = D − L− L2 − 3x2D2 − 2F,D = L+ S + P +H1 + 2F,L = 2x(D − L),S = D(D − S),P = x2(3D2 +D3) + 2FD,F = S2 +H2/2,

xS2 =(x2(2D +D2) + F

)(x2(2D +D2) + F − S2),

uH1 = T1,vH2 = T2,

u = 1 +D,x2v = x2(2D +D2) + F,

0 = PT1

(T (1), u2x, vx

),

0 = PT2

(T (2), u2x, vx

).

(22)

5.4 2-connected simple 4-regular planar maps

Let N(x) be the generating function 2-connected 4-regular simple maps.This case was not considered in [9], however it follows easily by removingloop networks in the previous system and we obtain

N = D − 3x2D2 − 2F,D = S + P +H1 + 2F,S = D(D − S),P = x2(3D2 +D3) + 2FD,F = S2 +H2/2,

xS2 =(x2(2D +D2) + F

)(x2(2D +D2) + F − S2),

uH1 = T1,vH2 = T2,

u = 1 +D,x2v = x2(2D +D2) + F,

0 = PT1

(T (1), u2x, vx

),

0 = PT2

(T (2), u2x, vx

).

(23)

6 Asymptotic enumeration

In this last section, we prove Theorems 1, 2 and 3. For the sake of clarity,we omit certain computational details in the proof of Theorem 3, whichcan be found in the accompanying Maple sessions. We First need thefollowing analytic lemma.

Lemma 4. Let f(x) be an algebraic generating function with non-negativecoefficients such that f(0) = 0. Further assume that f(x) admits a uniquedominant singularity ρ in the circle boarding the disk of convergence, anda Puiseux expansion as x → ρ− of the form:

f(x) = f0 + f2

(1− x

ρ

)+ f3

(1− x

ρ

)3/2

+O

((1− x

ρ

)2), (24)

14

with f0, f3 > 0 and f2 < 0. Then the coefficients of f(x) verify theasymptotic estimate

[xn]f(x) ∼ 3f34√π· n−5/2 · ρ−n, as n → ∞.

Proof. Since f(x) is algebraic and ρ is the unique singularity in the circleboarding the disk of convergence, by a classical compactness argument(see for instance the proof of [3, Theorem 2.19]), one can show that f(x)is analytic in a ∆-domain at ρ. We can apply the transfer theorem [4,Corollary VI.1] to the local representation (24), and deduce the estimateas claimed, using the relation Γ(−3/2) = 4

√π/3.

Proof of Theorem 1. The system of equations (20) shows thatC• = C•(x) is an analytic function of D = D(x). This implies in partic-ular that they both have the same singular behaviour. We first computethe equation satisfied by D from (20) minus the first equation. After elim-inating all the other variables, we obtain a polynomial in D and x withsix factors. The following three factors: 281474976710656, (1 +D)48 and(x+2)18 cannot be equal to zero. We can also discard two oother factors,one since its expansion at x = 0 has constant term −1/2, different fromzero, while the other admits an expansion of the form 1

2x6+ 1

4x7+O(x8),

which does not agree with the actual exponential generating function ofnetworks starting with 1

2x6 + 3

4x7. Hence the minimal polynomial of D

must be the remaining factor of degree 29

PD(D(x), x) =29∑

i=0

di(x) ·D(x)i. (25)

The discriminant pD(x) of PD(D,x) with respect to D has severalirreducible factors and we have to locate the one having the dominantsingularity as its root, which must be positive and less than 1. Once wediscard factors that do not have positive real roots less than 1, we have

pD(x) = f(x)g(x)2h(x)3,

where f, g have respective degrees 155 and 78, and h is the polynomialof degree 14 in the statement of Theorem 1. In order to rule out g andf , let us first recall that the dominant singularity of all labelled planargraphs is ρ1 ≈ 0.0367 [5]. The only candidate root for g is 0.00021,which can then be discarded because it is less than ρ1. The polynomialf has two candidate solutions: one is 0.026 and it is discarded for thesame reason as before; the other one is 0.86 and is discarded because it islarger than the singularity τ ≈ 0.24451 of 3-connected 4-regular graphs.Hence the dominant singularity ρ of D(x) (and of C(x) as argued above)is ρ ≈ 0.24377 the smallest positive root of h.

To compute the minimal polynomial of C•, we eliminate all the othervariables from (20) and obtain a polynomial equation in D, C• and x. Wecompute its resultant with PD with respect to D to obtain a polynomialin C• and x only. It has also six factors. We can discard four of them asthey trivially cannot be equal to zero, as before. Another factor admits

15

an expansion with constant term 7/8, different from zero, and can bediscarded too. Finally the minimal polynomial of C• is given by theremaining factor of degree 29

PC•(C•(x), x) =

29∑

i=0

ci(x) · C•(x)i. (26)

From PC• (C•, x) we compute the Puiseux expansion of C•(x) at x = ρassociated to the combinatorial branch, which turns out to be

C•(x) = C•0 + C•

2X2 + C•

3X3 +O(X4), (27)

where X =√

1− x/ρ, C0 ≈ 0.000057592, C2 ≈ −0.00098931 and C3 ≈0.0032877.

To obtain the estimate for the coefficients of C(x), we apply Lemma4 to (27) and divide the resulting estimate of the coefficients of C•(x)by n since there are n different ways to root a graph of size n at avertex. By integrating (27) we obtain the Puiseux expansion C(x) =C0 +C2X

2 +C3X3 +O(X4). However, the constant C0 is undetermined

after integration. The estimate for the coefficients of G(x) follows fromG(x) = exp(C(x)) and the corresponding Puiseux expansion

G(x) = G0 +G2X2 +G3X

3 +O(X4).

Since G3 = eC0C3, this coefficient cannot be determined either. As men-tioned after the statement of Theorem 1, we have estimated the constantg = G3/Γ(−3/2) from the first values of the coefficients gn. A similarsituation occurs in [8]. There, we circumvented it by using the so-called“dissymmetry theorem”. For this, one needs in particular the generatingfunction of unrooted 3-connected 4-regular planar graphs, which meansintegrating T1(u, v) with respect to u. We are not able to compute thisintegral; besides the fact that the size of the equation defining T1(u, v)is rather large, it defines a curves of genus 1, hence it does not admit arational parametrization.

Proof of Theorem 2. We will now compute an estimate the numberof 3-connected 4-regular planar graphs. By first plugging Equation (19)into the minimal polynomial of T1, we obtain the minimal polynomial ofT (1)(x, u, v). Setting then v = 0 and u = 1, and taking the root edgeinto account, it is a simple matter to check that we obtain a polynomialsatisfied by the generating function T •(x) of 3-connected 4-regular planargraphs rooted at a vertex, namely

4T •(x) = T (1)(x, 1, 0).

This polynomial is of the form

PT•(T •(x), x) =

8∑

i=0

ti(x) · T •(x)i,

where each ti(x) (i = 0, . . . , 8) is explicitly given in Appendix A.2.

16

Next, we compute the discriminant of PT• with respect to T •(x).It has five factors and we can discard two of them for trivial reasons.Another factor admits 0.0014891 as positive root, which is smaller thanρ1 ≈ 0.0367 and can be discarded. While another one has the positiveroot 0.53898, larger than 1/4 the dominant singularity of the generatingfunction of irreducible quadrangulations. It can then be discarded sincethe class of irreducible quadrangulations is contained in the class of sim-ple 3-connected quadrangulations, which is in bijection with the class of3-connected 4-regular planar graphs. So the dominant singularity must bethe the smallest positive root τ ≈ 0.24451 of the remaining factor, namely

3645x2 − 4752x + 944 = 0,

as claimed. The Puiseux expansion of T •(x) near τ is of the form

T •(x) = T •0 + T •

2 X2 + T •

3 X3 +O(X4), (28)

where X =√

1− x/τ , T •0 ≈ 0.000057426, T •

2 ≈ −0.00092862 and T •3 ≈

0.0028525. We conclude by applying Lemma 4 to (28) and dividing theresulting estimate by n.

To prove the estimate on the number of 2-connected 4-regular planargraphs, we proceed in the same way and thus omit certain details thatcan be found in the Maple sessions. Consider the system of equations(21) and eliminate all the other variables to obtain a single irreduciblebivariate polynomial equations in x and B•(x), as follows:

PB•(B•(x), x) =29∑

i=0

bi(x) ·B•(x)i.

The discriminant of PB• with respect to B•(x) has seven factors. Onlytwo of them have positive roots strictly smaller than one. The smallestsuch root of one of the factors is 0.013756, again smaller than ρ1. Thus,the dominant singularity β of B•(x) has to be the smallest positive rootof the remaining factor, which is the one claimed. The Puiseux expansionof B•(x) near β is of the form

B•(x) = B0 +B2X2 +B3X

3 +O(X4), (29)

where X =√

1− x/β, B0 ≈ 0.000057583, B2 ≈ −0.00098647 and B3 ≈0.0032669. We conclude again by applying Lemma 4 to (29) and dividingthe resulting estimate by n.

Proof of Theorem 3. The proof goes along the same lines as thetwo proofs above and we only briefly sketch it here. We refer the readerto the accompanying Maple sessions. First eliminate from (22) and (23)to obtain the minimal polynomials PM and PN satisfied by M(x) andN(x), respectively. We compute the discriminants of PM with respect toM(x) and of PN with respect to N(x), and find in each case the uniquefactor with a positive root. Then the smallest such root is the dominantsingularity. We conclude by applying Lemma 4 to the local expansions ofM(x) and N(x) near their respective dominant singularities.

17

7 Acknowledgements

The authors are very grateful to Thibaut Verron for suggesting the use ofevaluation and Lagrange interpolation to obtain the minimal polynomialsof T1(u, v) and T2(u, v). Part of this work was done while the second au-thor was a post-doctoral researcher under the direction of Manuel Kauers,at the Institute for Algebra of the Johannes Kepler Universitat Linz inAustria, and supported by the Special Research Program F50-04 Algo-rithmic and Enumerative Combinatorics of the Austrian Science Fund.

References

[1] E. A. Bender, Z. Gao, and N. C. Wormald. The number of labeled2-connected planar graphs. The Electronic Journal of Combinatorics,9(1):43, 2002.

[2] M. Bodirsky, M. Kang, M. Loffler, and C. McDiarmid. Random cubicplanar graphs. Random Structures & Algorithms, 30(1-2):78–94, 2007.

[3] M. Drmota. Random Trees: An Interplay Between Combinatorics andProbability. SpringerWienNewYork, 2010.

[4] P. Flajolet and R. Sedgewick. Analytic combinatorics. CambridgeUniversity Press, Cambridge, 2009.

[5] O. Gimenez and M. Noy. Asymptotic enumeration and limit laws ofplanar graphs. J. Amer. Math. Soc., 22(2):309–329, 2009.

[6] M. Kauers and P. Paule. The Concrete Tetrahedron. SpringerWien-NewYork, 2011.

[7] R. C. Mullin and P. J. Schellenberg. The enumeration of c-nets viaquadrangulations. Journal of Combinatorial Theory, 4:259–276, 1968.

[8] M. Noy, C. Requile, and J. Rue. Further results on random cubicplanar graphs. Random Structures and Algorithms (to appear).

[9] M. Noy, C. Requile, and J. Rue. Enumeration of labelled 4-regularplanar graphs. Proceedings of the London Mathematical Society,119(2):358–378, 2019.

18

A Appendix

We include in this appendix some of the polynomial equations needed inthe paper.

A.1 Minimal polynomials for quadrangulations

Coeffients of PB0(B0(z, w), z, w) =

∑2i=0 pi,0(z, w)B0(z, w)i:

• p2,0(z, w) = 27z2(wz − z − 1)2(wz − z + 1)3.

• p1,0(z, w) = −(wz−z−1)(16w8z12−144w7z12−32w7z11+560w6z12+16w7z10+224w6z11−1232w5z12−136w6z10−672w5z11+1680w4z12+40w6z9+544w5z10+1120w4z11−1456w3z12−296w5z9−1240w4z10−1120w3z11+784w2z12+204w5z8+816w4z9+1680w3z10+672w2z11−240wz12 − 1052w4z8 − 1104w3z9 − 1336w2z10 − 224wz11 + 32z12 −4w5z6+368w4z7+2184w3z8+776w2z9+576wz10+32z11−36w4z6−1440w3z7 − 2280w2z8 − 264wz9 − 104z10 − 10w4z5 + 444w3z6 +2112w2z7+1196wz8+32z9−w4z4−80w3z5−908w2z6−1376wz7−252z8 + 4w3z4 + 156w2z5 + 648wz6 + 336z7 + 2w3z3 − 10w2z4 −104wz5−144z6+20w2z3−156wz4+38z5+42wz3+163z4 +8wz2−208z3 − 2wz + 84z2 − 18z + 1).

• p0,0(z, w) = z(64w10z14−704w9z14−160w9z13+3456w8z14+32w9z12+1472w8z13 − 9984w7z14 − 272w8z12 − 6016w7z13 + 18816w6z14 +224w8z11+1264w7z12+14336w6z13−24192w5z14−16w8z10−2048w7z11−3920w6z12 − 21952w5z13 + 21504w4z14 + 408w7z10 + 7696w6z11 +8176w5z12+22400w4z13−13056w3z14−88w7z9−2536w6z10−15712w5z11−11312w4z12−15232w3z13+5184w2z14−8w7z8+928w6z9+7800w5z10+19120w4z11+10192w3z12+6656w2z13−1216wz14−60w6z8−3304w5z9−14120w4z10−14144w3z11−5744w2z12−1696wz13+128z14+8w6z7+664w5z8+5520w4z9+15816w3z10+6128w2z11+1840wz12+192z13−168w5z7−1676w4z8−4520w3z9−10808w2z10−1376wz11−256z12−3w5z6 − 168w4z7 + 1624w3z8 + 1408w2z9 + 4136wz10 + 112z11 +51w4z6+1712w3z7−388w2z8+232wz9−680z10+33w4z5−514w3z6−2584w2z7−296wz8−176z9+2w4z4+244w3z5+1458w2z6+1496wz7+140z8 + 2w3z4 − 774w2z5 − 1563wz6 − 296z7 − 4w3z3 + 170w2z4 +492wz5 + 571z6 − 54w2z3 − 98wz4 + 5z5 − 56wz3 − 12z4 − 7wz2 +162z3 + 4wz − 77z2 + 29z − 2).

Coeffients of PB1(B1(z, w), z, w) =

∑2i=0 pi,1(z, w)B1(z, w)i:

• p2,1(z, w) = 27z(wz − z − 1)2(wz − z + 1)3.

• p1,1(z, w) = −2w(wz − z − 1)(16w7z11 − 112w6z11 − 16w6z10 +336w5z11 + 24w6z9 + 96w5z10 − 560w4z11 − 176w5z9 − 240w4z10 +560w3z11 + 56w5z8 + 520w4z9 + 320w3z10 − 336w2z11 − 42w5z7 −232w4z8 − 800w3z9 − 240w2z10 + 112wz11 + 272w4z7 + 368w3z8 +680w2z9+96wz10−16z11−126w4z6−684w3z7−272w2z8−304wz9−16z10 + 2w4z5 + 548w3z6 + 840w2z7 + 88wz8 + 56z9 − 146w3z5 −888w2z6 − 506wz7 − 8z8 + 4w3z4 + 330w2z5 + 636wz6 + 120z7 −36w2z4 − 230wz5 − 170z6 +2w2z3+24wz4 +44z5 +w2z2 +20wz3 +8z4 + 10wz2 − 76z3 − 2wz + 79z2 − 18z + 1).

19

• p0,1(z, w) = −4w2z2(16w9z12 − 160w8z12 − 16w8z11 + 704w7z12 +24w8z10+160w7z11−1792w6z12−232w7z10−672w6z11+2912w5z12+56w7z9 +968w6z10 +1568w5z11 − 3136w4z12 − 15w7z8 − 408w6z9 −2280w5z10−2240w4z11+2240w3z12+171w6z8+1192w5z9+3320w4z10+2016w3z11−1024w2z12−45w6z7−722w5z8−1800w4z9−3064w3z10−1120w2z11 +272wz12 +2w6z6 +330w5z7 +1634w4z8 +1480w3z9 +1752w2z10 + 352wz11 − 32z12 − 67w5z6 − 850w4z7 − 2211w3z8 −616w2z9−568wz10−48z11+4w5z5+173w4z6+984w3z7+1799w2z8+88wz9+80z10−19w4z5−40w3z6−477w2z7−812wz8+8z9+2w4z4−94w3z5 − 278w2z6 + 22wz7 + 156z8 + w4z3 + 18w3z4 + 258w2z5 +307wz6+36z7+9w3z3−106w2z4−178wz5−97z6−2w3z2+55w2z3+195wz4 +29z5−17w2z2 −71wz3−109z4+w2z+wz2−21z3−wz+44z2 − 15z + 1).

Coeffients of PB∗

0(B∗

0 (z, w), z, w) =∑2

i=0 pi,2(z, w)B∗0(z, w)i:

• p2,2(z, w) = 27(wz − z + 1)2.

• p1,2(z, w) = 16w5z8−80w4z8−32w4z7+160w3z8+24w4z6+128w3z7−160w2z8−80w3z6−192w2z7+80wz8+24w3z5+96w2z6+128wz7−16z8 + 12w3z4 − 72w2z5 − 48wz6 − 32z7 + 96w2z4 + 72wz5 + 8z6 +12w2z3−228wz4 −24z5+2w2z2+120wz3+120z4 +8wz2 −132z3−4wz + 98z2 − 32z + 2.

• p0,2(z, w) = 4z3(8w5z7 − 40w4z7 − 16w4z6 + 80w3z7 + 8w4z5 +64w3z6 − 80w2z7 − 24w3z5 − 96w2z6 +40wz7 +20w3z4 +24w2z5 +64wz6 − 8z7 + 2w3z3 − 60w2z4 − 8wz5 − 16z6 + 29w2z3 + 60wz4 −2w2z2 − 64wz3 − 20z4 + 22wz2 + 33z3 + 2wz − 20z2 + 25z − 2).

A.2 Minimal polynomials for 3-connected planar

maps and graphs

Coeffients of PT1(T1(u, v), u, v) =

∑8i=0 ti,1(u, v)T1(u, v)

i:

• t8,1(u, v) = 1.

• t7,1(u, v) = −8u2 + 16uv + 2u+ 10.

• t6,1(u, v) = 28u4−112u3v+112u2v2−14u3+28u2v−69u2+152uv+68u+ 42.

• t5,1(u, v) = −56u6 +336u5v− 672u4v2 +448u3v3 +42u5 − 168u4v+168u3v2 + 204u4 − 900u3v + 984u2v2 − 408u3 + 840u2v − 92u2 +520uv + 352u + 96.

• t4,1(u, v) = 70u8 − 560u7v + 1680u6v2 − 2240u5v3 + 1120u4v4 −70u7 +420u6v−840u5v2+560u4v3−335u6 +2220u5v−4860u4v2+3520u3v3+1020u5−4200u4v+4320u3v2−170u4−988u3v+2728u2v2−1604u3 + 3280u2v + 394u2 + 776uv + 692u + 129.

• t3,1(u, v) = −56u10 + 560u9v − 2240u8v2 + 4480u7v3 − 4480u6v4 +1792u5v5+70u9−560u8v+1680u7v2−2240u6v3+1120u5v4+330u8−2920u7v + 9600u6v2 − 13920u5v3 + 7520u4v4 − 1360u7 + 8400u6v −17280u5v2 +11840u4v3 +760u6 − 1248u5v− 4416u4v2 +7744u3v3 +2896u5 − 11872u4v + 12256u3v2 − 2482u4 + 2756u3v + 2736u2v2 −1562u3 + 4224u2v + 1250u2 + 424uv + 550u+ 102.

20

• t2,1(u, v) = 28u12 − 336u11v + 1680u10v2 − 4480u9v3 + 6720u8v4 −5376u7v5 + 1792u6v6 − 42u11 + 420u10v − 1680u9v2 + 3360u8v3 −3360u7v4 +1344u6v5 − 195u10 +2160u9v− 9480u8v2 +20640u7v3 −22320u6v4+9600u5v5+1020u9−8400u8v+25920u7v2−35520u6v3+18240u5v4−970u8+4472u7v−3120u6v2−10144u5v3+12512u4v4−2584u7 +15936u6v− 33024u5v2+22976u4v3+4122u6 − 12600u5v+6024u4v2 + 5536u3v3 + 534u5 − 6960u4v + 9744u3v2 − 2829u4 +5636u3v + 392u2v2 + 722u3 + 1444u2v + 991u2 − 64uv + 104u+ 44.

• t1,1(u, v) = −8u14 +112u13v− 672u12v2 +2240u11v3 − 4480u10v4 +5376u9v5 − 3584u8v6 + 1024u7v7 + 14u13 − 168u12v + 840u11v2 −2240u10v3 + 3360u9v4 − 2688u8v5 + 896u7v6 + 64u12 − 852u11v +4680u10v2−13600u9v3+22080u8v4−19008u7v5+6784u6v6−408u11+4200u10v−17280u9v2+35520u8v3−36480u7v4+14976u6v5+548u10−3848u9v+8576u8v2−2944u7v3−11840u6v4+10880u5v5+1136u9−9376u8v+29280u7v2−40960u6v3+21632u5v4−2854u8+13828u7v−19608u6v2 +3504u5v3 +6464u4v4 +850u7 +1248u6v− 11112u5v2 +10560u4v3 + 1392u6 − 6412u5v + 6344u4v2 + 736u3v3 − 1312u5 +2216u4v+1160u3v2−356u4+2104u3v−504u2v2+626u3−312u2v−90u2 − 96uv − 40u+ 8.

• t0,1(u, v) = u2(u14 − 16u13v + 112u12v2 − 448u11v3 + 1120u10v4 −1792u9v5+1792u8v6−1024u7v7+256u6v8−2u13+28u12v−168u11v2+560u10v3−1120u9v4+1344u8v5−896u7v6+256u6v7−9u12+140u11v−924u10v2+3360u9v3−7280u8v4+9408u7v5−6720u6v6+2048u5v7+68u11−840u10v+4320u9v2−11840u8v3+18240u7v4−14976u6v5+5120u5v6 − 118u10 + 1092u9v − 3768u8v2 + 5344u7v3 − 672u6v4 −5568u5v5 + 3968u4v6 − 196u9 + 2032u8v − 8512u7v2 + 17984u6v3 −19136u5v4 +8192u4v5 +724u8 − 4760u7v+10848u6v2 − 8608u5v3 −1344u4v4 + 3328u3v5 − 514u7 + 1488u6v + 1368u5v2 − 7008u4v3 +4864u3v4 +58u6+352u5v− 1252u4v2 +80u3v3 +1232u2v4 +40u5 −96u4v−392u3v2+1072u2v3−8u4+292u2v2+128uv3+32uv2−16v2).

Coeffients of PT2(T2(u, v), u, v) =

∑8i=0 ti,2(u, v)T2(u, v)

i:

• t8,2(u, v) = (v + 1)5.

• t7,2(u, v) = 2v(v + 1)4(8uv + 8u− 4v + 1).

• t6,2(u, v) = 8v2(v+1)3(14u2v2+28u2v−14uv2+14u2−9uv+2v2+5u− 5v − 2).

• t5,2(u, v) = 8v3(v + 1)2(56u3v3 + 168u3v2 − 84u2v3 + 168u3v −129u2v2+24uv3+56u3−6u2v−48uv2+39u2−96uv+16v2−24u−6).

• t4,2(u, v) = 16v4(v + 1)(70u4v4 + 280u4v3 − 140u3v4 + 420u4v2 −340u3v3 + 60u2v4 + 280u4v − 180u3v2 − 90u2v3 + 70u4 + 100u3v −417u2v2 + 92uv3 + 80u3 − 324u2v + 94uv2 − 57u2 − 20uv + 25v2 −22u+ 22v + 2).

• t3,2(u, v) = 32v5(56u5v5+280u5v4−140u4v5+560u5v3−465u4v4+80u3v5+560u5v2−460u4v3−80u3v4+280u5v+10u4v2−788u3v3+208u2v4 + 56u5 + 240u4v − 1084u3v2 + 412u2v3 + 95u4 − 524u3v +169u2v2 + 96uv3 − 68u3 − 66u2v+ 182uv2 − 31u2 + 108uv + 19v2 +22u+ 26v + 8).

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• t2,2(u, v) = 64v6(28u6v5 +140u6v4 − 84u5v5 +280u6v3 − 270u5v4 +60u4v5+280u6v2−240u5v3−90u4v4+140u6v+60u5v2−672u4v3+232u3v4 + 28u6 + 180u5v − 876u4v2 + 440u3v3 + 66u5 − 396u4v +164u3v2 + 150u2v3 − 42u4 − 64u3v + 265u2v2 − 20u3 + 155u2v +12uv2 + 40u2 + 18uv + 12u+ 7v + 5).

• t1,2(u, v) = 128v7(8u7v5 + 40u7v4 − 28u6v5 + 80u7v3 − 87u6v4 +24u5v5 + 80u7v2 − 68u6v3 − 48u5v4 + 40u7v + 38u6v2 − 300u5v3 +128u4v4+8u7+72u6v−372u5v2+228u4v3+25u6−156u5v+67u4v2+112u3v3 − 12u5 − 38u4v + 168u3v2 − 5u4 + 76u3v + u2v2 + 20u3 −8u2v + 3u2 − 16uv − 6u+ 1).

• t0,2(u, v) = 256u2v8(u6v5+5u6v4−4u5v5+10u6v3−12u5v4+4u4v5+10u6v2 − 8u5v3 − 10u4v4 + 5u6v + 8u5v2 − 55u4v3 + 28u3v4 + u6 +12u5v − 65u4v2 + 46u3v3 + 4u5 − 25u4v + 8u3v2 + 33u2v3 − u4 −10u3v + 38u2v2 + 5u2v + 12uv2 + 4uv − v).

Coeffients of PT•(T •(x), x) =∑8

i=0 ti(x)T•(x)i:

• t8(x) = 16777216.

• t7(x) = 4194304(x + 1)(5− 4x).

• t6(x) = 7340032x4−3670016x3−18087936x2+17825792x+11010048.

• t5(x) = −1835008x6+1376256x5+6684672x4−13369344x3−3014656x2+11534336x + 3145728.

• t4(x) = 286720x8−286720x7−1372160x6+4177920x5−696320x4−6569984x3 + 1613824x2 + 2834432x + 528384.

• t3(x) = −28672x10 + 35840x9 + 168960x8 − 696320x7 + 389120x6 +1482752x5 − 1270784x4 − 799744x3 + 640000x2 + 281600x + 52224.

• t2(x) = 1792x12−2688x11−12480x10+65280x9−62080x8−165376x7+263808x6 +34176x5−181056x4+46208x3+63424x2+6656x+2816.

• t1(x) = −64x14 + 112x13 + 512x12 − 3264x11 + 4384x10 + 9088x9 −22832x8 +6800x7+11136x6 − 10496x5 − 2848x4+5008x3 − 720x2−320x + 64.

• t0(x) = x6(x2 + 4x − 1)(x8 − 6x7 + 16x6 − 2x5 − 94x4 + 178x3 −82x2 − 8x+ 8).

A.3 Minimal polynomials for simple maps

Coeffients of PM (M(x), x) =∑4

i=0 mi(x)M(x)i:

• m4(x) = (2x2 + 3x+ 3)6(x+ 1)4(2x+ 1)2(x− 1)2x6.

• m3(x) = −2x4(x−1)(64x14+480x13+1440x12+2496x11+276x10−11546x9−26420x8−19509x7+6393x6+19014x5+12975x4+4608x3+702x2 − 135x − 54)(x+ 1)3(2x2 + 3x+ 3)3.

• m2(x) = −x2(4608x26+39936x25+178688x24+520704x23+1094336x22+1543680x21+245408x20−8566240x19−32715326x18−59854300x17−53501976x16−7389020x15+36335841x14+52316608x13+51994151x12+41986758x11+22019337x10+1419738x9−10788681x8−14542164x7−13339809x6 − 9695160x5 − 5505759x4 − 2350134x3 − 725355x2 −148230x − 14823)(x + 1)2.

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• m1(x) = (x+1)(27648x31 +322560x30 +1847808x29 +6918144x28 +18257536x27+34084608x26+43349120x25+40257248x24+50673996x23+88945348x22+99083870x21+51494754x20+9311464x19−4802994x18−30341844x17−55300330x16−51227730x15−36294430x14−27913810x13−19958630x12 − 10165107x11 − 3664163x10 − 1317483x9 − 472215x8 +95764x7 + 263384x6 + 140418x5 + 22266x4 − 10179x3 − 6345x2 −783x + 459).

• m0(x) = −x6(x2 + 4x − 1)(34560x26 + 331776x25 + 1792512x24 +6458368x23+16625952x22+29597056x21+34923536x20+27157632x19+14306863x18+2833960x17−8516393x16−17003008x15−18205069x14−14628522x13−10556741x12−6840238x11−3542614x10−1345848x9−348274x8 − 14552x7 + 80947x6 + 78354x5 + 38619x4 + 8694x3 −1215x2 − 1512x − 459).

Coeffients of PN (N(x), x) =∑4

i=0 ni(x)N(x)i:

• n4(x) = (x2 + x+ 2)6(x+ 1)4(x− 1)2x3.

• n3(x) = −2x2(x− 1)(2x11 + 10x10 + 21x9 + 39x8 − 70x7 − 282x6 −329x5 + 112x4 + 271x3 + 67x2 + 25x + 6)(x+ 1)3(x2 + x+ 2)3.

• n2(x) = −x(18x22 +84x21 +344x20 +832x19 +1881x18 +2362x17 −213x16 − 22272x15 − 59887x14 − 60780x13 − 48821x12 + 2482x11 +70283x10 + 76870x9 + 64053x8 + 35032x7 − 9763x6 − 33312x5 −24499x4 − 8394x3 − 2308x2 − 328x − 48)(x+ 1)2.

• n1(x) = (x+1)(108x26 +828x25 +3798x24 +12498x23 +27832x22 +44416x21+45122x20+52780x19+83480x18+38444x17+52052x16+63060x15−24937x14+18259x13−10897x12−78617x11−16174x10−16450x9 − 26700x8 − 984x7 + 3531x6 + 3159x5 + 361x4 − 641x3 +144x2 − 48x + 8).

• n0(x) = −x6(x2 + 4x− 1)(135x21 + 756x20 + 3843x19 + 10936x18 +25151x17 + 30928x16 + 31103x15 + 27536x14 + 3310x13 − 7148x12 −10618x11 − 21620x10 − 13409x9 − 5500x8 − 4305x7 − 60x6 +965x5 +604x4 + 201x3 − 104x2 + 8x− 8).

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